Wireless Internet of Things (IoT) devices will be deployed to enable applications such as sensing and actuation. These devices are typically resource-constrained and are unable to perform resource-intensive computations. Therefore, these jobs need to be offloaded to resource-rich nodes at the edge of the IoT network for execution. However, the timeliness and correctness of edge nodes may not be trusted (such as during high network load or attack). In this paper, we look at the applicability of trust for successful offloading. Traditionally, trust is computed at the application level, with suitable mechanisms to adjust for factors such as recency. However, these do not work well in IoT networks due to resource constraints. We propose a novel device called Trust Tracker (denoted by Σ) that provides higher-level applications with up-to-date trust information of the resource-rich nodes. We prove impossibility results regarding computation offloading and show that Σ is necessary and sufficient for correct offloading. We show that, Σ cannot be implemented even in a synchronous network and we compute the probability of offloading to a bad node, which we show to be negligible when a majority of nodes are correct. We perform a small-scale deployment to demonstrate our approach.